Like any full contact martial art, injuries are commonplace in Kendo.
But, most of those are not the result of a shinai hit, although these aren't heard of.
The beginners usually have problems with their feet and hands. People that have thin skin on the soles of their feet often get blisters (kendo is practiced barefoot, and most Dojos have wooden floors), usually in the area of thumbs and the top of the foot. The skin will gradually gain thickness, but a lot of kendokas wear bandages or specialized footwear called tabi. It's similar with hands – in the first few months of training, kendokas hold the shinai grip too tightly. The result – blisters. This should be mended with bandages in the short-term, but the problem should be corrected as soon as possible. As in all injuries that break the skin, watch out for infection. Always clean and treat the wound, especially on the feet.
Muscle fatigue is, like in any other sports, an often occurrence. The hands and shoulders suffer the most, but legs are no exception. You shouldn't push yourselves too hard, because pain is always counter-productive. If you fell used up and soar, take it down a notch. The way of the sword won't go away while you recuperate.
When practitioners start wearing armor, injuries start to shift towards bruises. The unprotected parts of arms are mostly in the line of fire, and elbows get the most of it. A shinai hit on a bone can be very painful, and if the same place is hit several times, protection like pads can be considered. Don't shy away from this if others don't wear them – if you feel better with them, you will perform better in the Dojo.
If you train in a humid and/or hot space, take a lot of fluids and regular breaks, unless you want to faint.
But, the biggest danger of Kendo is carelessness. Always carefully put on your bogu, or the armor, and check out all of it once you do. If something isn't feeling right, or loose, call your teacher. All malfunctions and tears should be repaired as soon as possible. In the same time, don't tie the knots too tight, especially those on the men. They can create pressure on the arteries in the neck, and diminish the blood flow to the head. Not a good idea. The same applies for shinai – if it's showing cracks and splinters, it's time for a new one. Be focused, be mindful, be safe, have fun and enjoy Kendo.